Question

Which data set is the most clustered around its mean?
4, 10, 8, 6
11, 3, 10, 4
2, 9 ,13, 4
11, 2, 9, 6

Answer 1

To determine which dataset is most clustered around its mean, we can calculate the mean and the standard deviation for each dataset. A smaller standard deviation indicates that the data points are closer to the mean (more clustered).

Let's analyze each dataset:

1. **Dataset 1: 4, 10, 8, 6**
- Mean = (4 + 10 + 8 + 6) / 4 = 7
- Variance = [(4-7)² + (10-7)² + (8-7)² + (6-7)²] / 4
= [(9) + (9) + (1) + (1)] / 4 = 20 / 4 = 5
- Standard Deviation = √5 ≈ 2.24

2. **Dataset 2: 11, 3, 10, 4**
- Mean = (11 + 3 + 10 + 4) / 4 = 7
- Variance = [(11-7)² + (3-7)² + (10-7)² + (4-7)²] / 4
= [(16) + (16) + (9) + (9)] / 4 = 50 / 4 = 12.5
- Standard Deviation = √12.5 ≈ 3.54

3. **Dataset 3: 2, 9, 13, 4**
- Mean = (2 + 9 + 13 + 4) / 4 = 7
- Variance = [(2-7)² + (9-7)² + (13-7)² + (4-7)²] / 4
= [(25) + (4) + (36) + (9)] / 4 = 74 / 4 = 18.5
- Standard Deviation = √18.5 ≈ 4.30

4. **Dataset 4: 11, 2, 9, 6**
- Mean = (11 + 2 + 9 + 6) / 4 = 7
- Variance = [(11-7)² + (2-7)² + (9-7)² + (6-7)²] / 4
= [(16) + (25) + (4) + (1)] / 4 = 46 / 4 = 11.5
- Standard Deviation = √11.5 ≈ 3.39

Now let's summarize the standard deviations:

- Dataset 1: SD ≈ 2.24
- Dataset 2: SD ≈ 3.54
- Dataset 3: SD ≈ 4.30
- Dataset 4: SD ≈ 3.39

The dataset with the smallest standard deviation is **Dataset 1: 4, 10, 8, 6**, which means it is the most clustered around its mean.